L(s) = 1 | + 6.40i·3-s + 47.0·5-s − 32.2·7-s + 40.0·9-s + 32.3·11-s + 258. i·13-s + 301. i·15-s + 35.6·17-s + (147. − 329. i)19-s − 206. i·21-s + 5.36·23-s + 1.59e3·25-s + 774. i·27-s − 838. i·29-s + 704. i·31-s + ⋯ |
L(s) = 1 | + 0.711i·3-s + 1.88·5-s − 0.657·7-s + 0.494·9-s + 0.267·11-s + 1.53i·13-s + 1.33i·15-s + 0.123·17-s + (0.409 − 0.912i)19-s − 0.467i·21-s + 0.0101·23-s + 2.54·25-s + 1.06i·27-s − 0.996i·29-s + 0.733i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.865881034\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.865881034\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-147. + 329. i)T \) |
good | 3 | \( 1 - 6.40iT - 81T^{2} \) |
| 5 | \( 1 - 47.0T + 625T^{2} \) |
| 7 | \( 1 + 32.2T + 2.40e3T^{2} \) |
| 11 | \( 1 - 32.3T + 1.46e4T^{2} \) |
| 13 | \( 1 - 258. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 35.6T + 8.35e4T^{2} \) |
| 23 | \( 1 - 5.36T + 2.79e5T^{2} \) |
| 29 | \( 1 + 838. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 704. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.26e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.36e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.69e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 4.03e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 2.00e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 4.43e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.13e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.95e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 6.54e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.33e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.87e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 9.80e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 6.93e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.03e4iT - 8.85e7T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.96451515638934343890366388333, −10.07021806674807368760943904921, −9.409165068306034739054374291533, −9.038709376448995686956760484095, −7.02299646414165081918669532565, −6.33222295082609910320698646026, −5.24331085837105596692354320223, −4.16892113068182824781966262429, −2.62984633258144794209783256278, −1.42066764849123971167024584508,
0.966395800945692440362159699223, 2.02083951936309318650136146929, 3.26522705558441431517794009719, 5.21696590300138674341034471655, 6.03661999224741166129305482875, 6.79335910212705795654435257918, 7.948602097836221787811585604783, 9.225555379216639919362340541632, 10.03537992650906913555992519887, 10.52002819488111234027576452786